Numbers Divisible by 9



Table below shows a list of multiples of 9 and the sum of their digits. We realised that, the sum of the digits is also multiple of 9.

Multiples of 9 Sum of the digits
9 9
18 1 + 8 = 9
27 2 + 7 = 9
36 3 + 6 = 9
45 4 + 5 = 9
54 5 + 4 = 9
63 6 + 3 = 9
72 7 + 2 = 9
81 8 + 1 = 9
90 9 + 0 = 9
99 9 + 9 = 18 1 + 8 = 9
108 1 + 0 + 8 = 9
117 1 + 1 + 7 = 9
126 1 + 2 + 6 = 9
135 1 + 3 + 5 = 9
144 1 + 4 + 4 = 9
153 1 + 5 + 3 = 9
162 1 + 6 + 2 = 9
171 1 + 7 + 1 = 9
180 1 + 8 + 0 = 9
189 1 + 8 + 9 = 18 1 + 8 = 9
198 1 + 9 + 8 = 18 1 + 8 = 9
207 2 + 0 + 7 = 9
216 2 + 1 + 6 = 9
225 2 + 2 + 5 = 9
234 2 + 3 + 4 = 9
243 2 + 4 + 3 = 9
252 2 + 5 + 2 = 9
261 2 + 6 + 1 = 9

Therefore, we can conclude that
If the sum of the digits of a number is divisible by 9, the number must also divisible by 9
or
If the sum of the digits of a number is a multiple of 9, the number must also a multiple of 9.
For example:
Is 972635 a multiple of 9?

Sum of the digits:
9 + 7 + 2 + 6 + 3 + 5 = 32

32 is not multiple of 9, hence 972635 is not a multiple of 9.

Let's see another example:
Is 683746533 divisible by 9?

Sum of the digits:
6 + 8 + 3 + 7 + 4 + 6 + 5 + 3 + 3 = 45

45 is divisible by 9, hence 683746533 is divisible by 9

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